A VPS is a virtual private server. People use these to increase the size of their farm. Its basically you controlling another computer from your computer. And through an app you can control that computer to run more bots - Proxies make you able to change your ip adress, so it looks like your accounts are coming from different places. Residential proxies are proxies that dont come from data-centers, eg your own house. dedicated ones are the most common ones being sold and are much cheaper(these are the ones that come from datacenters) However recently jagex can detect if you are using datacenter proxies and have been clamping down on data-center proxies and locking account that are coming from these proxies. However if you use a residential one u wont recieve locks but are much more expensive - probs should look at a rotating one.
Hope this helps a bit